Analogously to the formulation developed in [1]} for the one-liquid case, equation (REF ) encompasses the effects of both capillary- and gravity-driven drainage on the steady state film thicknesses.
In this work, we will focus on the case where capillary effects prevail over gravity.
In equation (REF ), viscous stresses will be of the same order as capillary ones if \(\varepsilon = Ca^{2/3}\) , which is the same scaling obtained in the one-liquid case by [2]}.
Finally, at leading order, the equations satisfied by the thicknesses are
\(\frac{\partial }{\partial \hat{z}}\left[\hat{h}_i + \frac{\partial ^3 \hat{h}_1}{\partial \hat{z}^3}\hat{F}_{i1} +\Sigma \, \frac{\partial ^3 \hat{h}_2}{\partial \hat{z}^3}\left(\hat{F}_{i1} + \hat{F}_{i2}\right)\right] = 0.\) 
